Respuesta :
Answer:
a) [tex]2259.92-2.201\frac{35.569}{\sqrt{12}}=2237.32[/tex]
[tex]2259.92+2.201\frac{35.569}{\sqrt{12}}=2282.52[/tex]
So on this case the 95% confidence interval would be given by (2237.32;2282.52)
b) 1) Randomization we assume that the sample is selected from a random sample
2) Independence we assume that each observation is independent from the other ones
3) Sample size condition the sampe needs to be less than 10% of the population size.
c) For this case we can conclude with 95% of confidence that the true mean of compressive strength of concrete is between 2237.32 and 2282.52
Step-by-step explanation:
Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
[tex]\bar X[/tex] represent the sample mean for the sample
[tex]\mu[/tex] population mean (variable of interest)
s represent the sample standard deviation
n represent the sample size
Part a
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (1)
In order to calculate the mean and the sample deviation we can use the following formulas:
[tex]\bar X= \sum_{i=1}^n \frac{x_i}{n}[/tex] (2)
[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar X)}{n-1}}[/tex] (3)
The mean calculated for this case is [tex]\bar X=2259.92[/tex]
The sample deviation calculated [tex]s=35.569[/tex]
In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:
[tex]df=n-1=12-1=11[/tex]
Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,11)".And we see that [tex]t_{\alpha/2}=2.201[/tex]
Now we have everything in order to replace into formula (1):
[tex]2259.92-2.201\frac{35.569}{\sqrt{12}}=2237.32[/tex]
[tex]2259.92+2.201\frac{35.569}{\sqrt{12}}=2282.52[/tex]
So on this case the 95% confidence interval would be given by (2237.32;2282.52)
Part b
1) Randomization we assume that the sample is selected from a random sample
2) Independence we assume that each observation is independent from the other ones
3) Sample size condition the sampe needs to be less than 10% of the population size.
Part c
For this case we can conclude with 95% of confidence that the true mean of compressive strength of concrete is between 2237.32 and 2282.52
Using the t-distribution, we have that:
a) The 95% two-sided confidence interval on the mean strength is (2237, 2283).
b) We have to assume that the underlying distribution is normal.
c) It means that we are 95% sure that the average compressive strength of concrete of all specimens is between 2237 and 2283.
Item a:
The sample size is of [tex]n = 12[/tex]
Using a calculator:
- The sample mean is [tex]\overline{x} = 2260[/tex]
- The sample standard deviation is [tex]s = 35.57[/tex]
The confidence interval is:
[tex]\overline{x} \pm t\frac{s}{\sqrt{n}}[/tex]
The critical value, using a t-distribution calculator, for a two-tailed 95% confidence interval, with 12 - 1 = 11 df, is t = 2.201.
Then, the interval is:
[tex]\overline{x} - t\frac{s}{\sqrt{n}} = 2260 - 2.201\frac{35.57}{\sqrt{12}} = 2237[/tex]
[tex]\overline{x} + t\frac{s}{\sqrt{n}} = 2260 + 2.201\frac{35.57}{\sqrt{12}} = 2283[/tex]
The 95% two-sided confidence interval on the mean strength is (2237, 2283).
Item b:
We have to assume that the underlying distribution, that is, the compressive strength of concrete of all specimens, is normal.
Item c:
It means that we are 95% sure that the average compressive strength of concrete of all specimens is between 2237 and 2283.
A similar problem is given at https://brainly.com/question/15180581
